Applicability of Rule of Mixtures to Estimate Effective Properties of Nanocomposite Materials

Abstract

A rule of mixtures is employed and modified to examine the effective elastic properties of a unidirectional composite lamina reinforced with nanostructure-hybrid fibers. Such fiber system is designated when nanostructure such as nanowires or carbon nanotubes are radially grown on the surface of primary fiber. When combines with matrix, a complex three-phase composite with enhanced elastic properties is expected. Herein, the applicability of this simple micromechanics method to reliably estimate the effective properties of such advanced novel composite material is assessed. The results demonstrated that the method is capable of modeling the effects on elastic properties of a composite due to the presence of nanostructure. However, in light of published experimental data and other micromechanics results, the proposed method is found to be at best, applicable for a composite that has a very low fiber volume fraction only except when an axial Young’ modulus is predicted.

Appendix

The details of mathematical modeling of the ROM model modified for a three orthotropic phase lamina that yields final expressions for the four effective properties shown in Eq. (2) are derived here. Referring to Fig. 2, the constitutive equations for the effective composite or lamina [18] are given as

$$ \begin{aligned} & \varepsilon_{3} = \sigma_{3} /E_{3}^{*} {-}v_{13}^{*} \sigma_{1} /E_{1}^{*} , \\ & \varepsilon_{1} = \sigma_{1} /E_{1}^{*} {-}v_{31}^{*} \sigma_{3} /E_{3}^{*} , \\ & \gamma_{31} = \tau_{31} /G_{31}^{*} \\ \end{aligned} $$

(3)

where E^*₃—effective axial Young’s modulus, E^*₁—effective transverse Young’s modulus, G^*₃₁  =  G^*₃₁—effective shear modulus, σ_3,1—normal stresses in axial and transverse direction experienced by the lamina respectively, ε_3,1—normal strain in axial and transverse direction respectively, τ₃₁—shear stress, γ₃₁—shear strain, ν^*₃₁—effective major Poisson’s ratio and ν^*₁₃—effective minor Poisson’s ratio obeying the following general symmetry

$$ E_{3} \nu_{13} = E_{1} \nu_{31} . $$

(4)

The volume fraction of every constituent is given as follows

$$ \upsilon_{f,i,m} = w_{f,i,m} /w $$

(5)

and as before, $\upsilon_{f} + \upsilon_{i} + \upsilon_{m} = \upsilon = 1$ where υ designates total volume fraction and w is the total width of a lamina and with subscript f, i or m, they represent the quantity for fiber, interphase and matrix phase respectively. Finally, the constitutive equations for every constituent, which are similar to the relations given for lamina in Eq. (3) but have been rearranged with the aid of Eq. (4), can be written as

$$ \begin{aligned} & \varepsilon_{3}^{f,i,m} = (\sigma_{3}^{f,i,m} {-} \, v_{31}^{f,i,m} \sigma_{1}^{f,i,m} )/E_{3}^{f,i,m} , \\ & \varepsilon_{1}^{f,i,m} = (\sigma_{1}^{f,i,m} {-} \, v_{13}^{f,i,m} \sigma_{3}^{f,i,m} )/E_{1}^{f,i,m} , \\ & \gamma_{31}^{f,i,m} = \tau_{31}^{f,i,m} /G_{31}^{f,i,m} . \\ \end{aligned} $$

(6)

Next, we shall define the states of stresses and strains of the constituents under specific loading conditions applied to the lamina. When a lamina is axially loaded in tension, the iso-strain state condition requires every constituent or strip to experience uniform strain as the lamina, which we have

$$ \varepsilon_{3} = \varepsilon_{3}^{f} = \varepsilon_{3}^{i} = \varepsilon_{3}^{m} $$

(7)

and the apparent resultant stress of a lamina is distributed accordingly between the constituents, which is given as follow

$$ \sigma_{3} w = \sigma_{3}^{f} w_{f} + \sigma_{3}^{i} w_{i} + \sigma_{3}^{m} w_{m} . $$

(8)

In transverse loading condition, iso-stress condition requires all strips to experience equal amount of apparent stress experienced by the lamina, which gives us

$$ \sigma_{1} = \sigma_{1}^{f} = \sigma_{1}^{i} = \sigma_{1}^{m} . $$

(9)

Furthermore, the apparent elongation of a lamina is basically the sum of fiber, interphase and matrix strip’s elongation, e.g. $\Delta w = \Delta w_{f} + \Delta w_{i} + \Delta w_{m}$. Introducing transverse strains for the lamina and also for every constituent: $\varepsilon_{1} = \Delta w/w$, $\varepsilon_{1}^{f} = \Delta w^{f} /w^{f}$, $\varepsilon_{1}^{i} = \Delta w^{i} /w^{i}$ and $\varepsilon_{1}^{m} = \Delta w^{m} /w^{m}$, the apparent elongation of a lamina can now be written as

$$ \varepsilon_{1} w = \varepsilon_{1}^{f} w^{f} + \varepsilon_{1}^{i} w^{i} + \varepsilon_{1}^{m} w^{m} . $$

(10)

Finally, similar assumptions employed in the transverse loading condition are applicable to shear loading case, in which the shear stresses of lamina and its constituents are found to be

$$ \tau_{31} = \tau_{31}^{f} = \tau_{31}^{i} = \tau_{31}^{m} $$

(11)

while the shear strain of a lamina are defined as

$$ \gamma_{31} w = \gamma_{31}^{f} w^{f} + \gamma_{31}^{i} w^{i} + \gamma_{31}^{m} w^{m} . $$

(12)

Once the states of stresses and strains are properly defined, we now proceed to the determination of effective properties of the lamina. First, we wish to determine the expression for an effective axial Young’s modulus. Considering Eqs. (7), (9) and (11), the constitutive equations for every constituent given in Eq. (6) can be reduced to the following form

$$ \begin{aligned} & \varepsilon_{3} = (\sigma_{3}^{f} {-}v_{31}^{f} \sigma_{1} )/E_{3}^{f} ,\varepsilon_{3} = (\sigma_{3}^{i} {-}v_{31}^{i} \sigma_{1} )/E_{3}^{i} ,\varepsilon_{3} = (\sigma_{3}^{m} {-}v_{31}^{m} \sigma_{1} )/E_{3}^{m} , \\ & \varepsilon_{1}^{f} = (\sigma_{1} {-}v_{13}^{f} \sigma_{3}^{f} )/E_{1}^{f} ,\varepsilon_{1}^{i} = (\sigma_{1} {-}v_{13}^{i} \sigma_{3}^{i} )/E_{1}^{i} ,\varepsilon_{1}^{m} = (\sigma_{1} {-}v_{13}^{m} \sigma_{3}^{m} )/E_{1}^{m} , \\ & \gamma_{31}^{f} = \tau_{31} /G_{31}^{f} ,\gamma_{31}^{i} = \tau_{31} /G_{31}^{i} ,\gamma_{31}^{m} = \tau_{31} /G_{31}^{m} . \\ \end{aligned} $$

(13)

The three relations in the first row of Eq. (13) can be rewritten to define the axial stress experienced by every strip, e.g.

$$ \sigma_{3}^{f} = \varepsilon_{3} E_{3}^{f} + v_{31}^{f} \sigma_{1} ,\sigma_{3}^{i} = \varepsilon_{3} E_{3}^{i} + v_{31}^{i} \sigma_{1} ,\sigma_{3}^{m} = \varepsilon_{3} E_{3}^{m} + v_{31}^{m} \sigma_{1} . $$

(14)

Next, Eq. (8) can be rewritten with the aid of Eq. (5) giving us

$$ \sigma_{3} = \sigma_{3}^{f} \upsilon_{f} + \sigma_{3}^{i} \upsilon_{i} + \sigma_{3}^{m} \upsilon_{m} . $$

(15)

It is in our intention to express ε₃ in terms of σ₃ and σ₁ and this is achieved by substituting Eq. (14) into (15), which we have

$$ \begin{aligned} \varepsilon_{3} & = \sigma_{3} /\left( {E_{3}^{f} \upsilon_{f} + E_{3}^{i} \upsilon_{i} + E_{3}^{m} \upsilon_{m} } \right) \\ & \quad {-}\left[ {\left( {v_{31}^{f} \upsilon_{f} + v_{31}^{i} \upsilon_{i} + v_{31}^{m} \upsilon_{m} } \right)/\left( {E_{3}^{f} \upsilon_{f} + E_{3}^{i} \upsilon_{i} + E_{3}^{m} \upsilon_{m} } \right)} \right]\sigma_{1} . \\ \end{aligned} $$

(16)

Here, we are about to get the desired expression of an effective axial Young’s modulus. Equating Eq. (16) with the first expression of Eq. (3), the following relations can be obtained

$$ E_{3}^{*} = E_{3}^{f} \upsilon_{f} + E_{3}^{i} \upsilon_{i} + E_{3}^{m} \upsilon_{m} , $$

(17)

$$ v_{13}^{*} /E_{1}^{*} = \left( {v_{31}^{f} \upsilon_{f} + v_{31}^{i} \upsilon_{i} + v_{31}^{m} \upsilon_{m} } \right)/\left( {E_{3}^{f} \upsilon_{f} + E_{3}^{i} \upsilon_{i} + E_{3}^{m} \upsilon_{m} } \right). $$

(18)

As we can see, Eq. (17) defines the effective axial Young’s modulus for our lamina.

Next, we wish to determine the expression for an effective transverse Young’s modulus. Using Eq. (5), we can rewrite Eq. (10) as

$$ \varepsilon_{1} = \varepsilon_{1}^{f} \upsilon_{f} + \varepsilon_{1}^{i} \upsilon_{i} + \varepsilon_{1}^{m} \upsilon_{m} . $$

(19)

Substituting the three equations shown in the second row of Eq. (13) into (19) and taking into account Eq. (14) together with the first row relations of Eq. (13) as well as both Eqs. (17) and (18), we will arrive to a relation that expresses ε₁ in terms of σ₁ and σ₃ and comparing that newly obtained relation with the second expression of Eq. (3), we receive

$$ \begin{aligned} 1/E_{1}^{*} & = \upsilon_{f} /E_{1}^{f} + \upsilon_{i} /E_{1}^{i} + \upsilon_{m} /E_{1}^{m} \\ & \quad {-}\left[ {\left( {E_{3}^{f} v_{31}^{i} + E_{3}^{i} v_{31}^{f} } \right)\left( {E_{1}^{f} v_{13}^{i} + E_{1}^{i} v_{13}^{f} } \right)E_{1}^{m} \upsilon_{f} \upsilon_{i} } \right]/\left[ {E_{1}^{f} E_{1}^{i} E_{1}^{m} \left( {E_{3}^{f} \upsilon_{f} + E_{3}^{i} \upsilon_{i} + E_{3}^{m} \upsilon_{m} } \right)} \right] \\ & \quad {-}\left[ {\left( {E_{3}^{f} v_{31}^{m} + E_{3}^{m} v_{31}^{f} } \right)\left( {E_{1}^{f} v_{13}^{m} + E_{1}^{m} v_{13}^{f} } \right)E_{1}^{i} \upsilon_{f} \upsilon_{m} } \right]/\left[ {E_{1}^{f} E_{1}^{i} E_{1}^{m} \left( {E_{3}^{f} \upsilon_{f} + E_{3}^{i} \upsilon_{i} + E_{3}^{m} \upsilon_{m} } \right)} \right] \\ & \quad {-}\left[ {\left( {E_{3}^{i} v_{31}^{m} + E_{3}^{m} v_{31}^{i} } \right)\left( {E_{1}^{i} v_{13}^{m} + E_{1}^{m} v_{13}^{i} } \right)E_{1}^{f} \upsilon_{i} \upsilon_{m} } \right]/\left[ {E_{1}^{f} E_{1}^{i} E_{1}^{m} \left( {E_{3}^{f} \upsilon_{f} + E_{3}^{i} \upsilon_{i} + E_{3}^{m} \upsilon_{m} } \right)} \right] \\ \end{aligned} $$

(20)

and

$$ v_{31}^{*} /E_{3}^{*} = \left( {v_{31}^{f} \upsilon_{f} + v_{31}^{i} \upsilon_{i} + v_{31}^{m} \upsilon_{m} } \right)/\left( {E_{3}^{f} \upsilon_{f} + E_{3}^{i} \upsilon_{i} + E_{3}^{m} \upsilon_{m} } \right). $$

(21)

Here, Eq. (20) represents the needed expression for an effective transverse Young’s modulus. Based on our computations with entire possible range of fiber volume fraction, the results indicated that the last three terms on the right-hand side of Eq. (20) can safely be ignored and thus, giving us the simplified and familiar expression of the said modulus such as shown in Eq. (2) earlier.

Next, we shall define the expression for an effective major Poisson’s ratio and this is easily received by substituting Eqs. (17) into (21), which we have

$$ v_{31}^{*} = v_{31}^{f} \upsilon_{f} + v_{31}^{i} \upsilon_{i} + v_{31}^{m} \upsilon_{m} . $$

(22)

On the other hand, to determine the expression for effective shear modulus, we first substitute Eqs. (5) into (12) and obtain

$$ \gamma_{31} = \gamma_{31}^{f} \upsilon_{f} + \gamma_{31}^{i} \upsilon_{i} + \gamma_{31}^{m} \upsilon_{m} . $$

(23)

Then, having the last relations of Eq. (13) substituted into (23) and subsequently equating it with last row expression of Eq. (3), we receive

$$ 1/G_{31}^{*} = \upsilon_{f} /G_{31}^{f} + \upsilon_{i} /G_{31}^{i} + \upsilon_{m} /G_{31}^{m} . $$

(24)

Thus, Eqs. (22) and (24) completes our tasks in determining the expressions for all effective moduli of the lamina.